The technique introduced in Siegel's 1926 letter to Mordell [1]} to prove the finiteness of integral points on hyperelliptic curvesBy this common and incorrect abbreviation we really mean integral solutions of \(y^2 = f(x)\) . Siegel's 1926 proof does not control integral solutions of e.g. \(y^3 + x\cdot y = x^4\) , or, said another way, integral points with respect to an effective divisor containing no nonzero divisor symmetric under the hyperelliptic involution. Baker effectivized Siegel's 1926 argument and thus gave an effective finiteness proof in the \(y^2 = f(x)\)  cases, but he did not effectivize Siegel's 1929 finiteness theorem in the case of hyperelliptic curves. can be summarized as: 2-descent on the curve, then 3-descent on \(\mathbb {G}_m\) . From this summary the intuitive guess for the bound that the argument "should" produce is of course something of shape \(2^{\mathop {\mathrm {rank}}}\cdot 3^{\text{$\#$ of prime factors of the discriminant}}\) . However a consultation of the literature yields worse estimates.To our knowledge the best bounds in the literature are the 2006 estimate of Bombieri-Gubler [2]} and the 1986 estimate of Evertse-Silverman [3]}. In both bounds the "expected" \(2^{\mathop {\mathrm {rank}}}\)  factor is replaced by a power of the size of the 2-part of the class group of an extension gotten by adjoining three Weierstrass points.
